L(s) = 1 | + (−0.615 − 0.788i)2-s + (1.44 − 1.39i)3-s + (−0.241 + 0.970i)4-s + (1.24 − 0.779i)5-s + (−1.98 − 0.278i)6-s + (0.930 + 0.414i)7-s + (0.913 − 0.406i)8-s + (0.0352 − 1.00i)9-s + (−1.38 − 0.502i)10-s + (2.19 + 2.48i)11-s + (1.00 + 1.73i)12-s + (4.07 − 2.16i)13-s + (−0.246 − 0.988i)14-s + (0.712 − 2.85i)15-s + (−0.882 − 0.469i)16-s + (−0.137 − 3.93i)17-s + ⋯ |
L(s) = 1 | + (−0.435 − 0.557i)2-s + (0.831 − 0.803i)3-s + (−0.120 + 0.485i)4-s + (0.557 − 0.348i)5-s + (−0.809 − 0.113i)6-s + (0.351 + 0.156i)7-s + (0.322 − 0.143i)8-s + (0.0117 − 0.336i)9-s + (−0.436 − 0.159i)10-s + (0.662 + 0.748i)11-s + (0.289 + 0.500i)12-s + (1.12 − 0.600i)13-s + (−0.0658 − 0.264i)14-s + (0.183 − 0.737i)15-s + (−0.220 − 0.117i)16-s + (−0.0333 − 0.953i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.27247 - 1.07341i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27247 - 1.07341i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.615 + 0.788i)T \) |
| 11 | \( 1 + (-2.19 - 2.48i)T \) |
| 19 | \( 1 + (3.36 + 2.76i)T \) |
good | 3 | \( 1 + (-1.44 + 1.39i)T + (0.104 - 2.99i)T^{2} \) |
| 5 | \( 1 + (-1.24 + 0.779i)T + (2.19 - 4.49i)T^{2} \) |
| 7 | \( 1 + (-0.930 - 0.414i)T + (4.68 + 5.20i)T^{2} \) |
| 13 | \( 1 + (-4.07 + 2.16i)T + (7.26 - 10.7i)T^{2} \) |
| 17 | \( 1 + (0.137 + 3.93i)T + (-16.9 + 1.18i)T^{2} \) |
| 23 | \( 1 + (4.68 - 3.93i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (4.36 - 1.25i)T + (24.5 - 15.3i)T^{2} \) |
| 31 | \( 1 + (-4.79 + 5.32i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (3.55 - 2.58i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (0.264 - 0.255i)T + (1.43 - 40.9i)T^{2} \) |
| 43 | \( 1 + (7.84 + 6.58i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-6.29 - 9.33i)T + (-17.6 + 43.5i)T^{2} \) |
| 53 | \( 1 + (8.91 + 5.57i)T + (23.2 + 47.6i)T^{2} \) |
| 59 | \( 1 + (0.405 - 0.600i)T + (-22.1 - 54.7i)T^{2} \) |
| 61 | \( 1 + (0.707 + 1.75i)T + (-43.8 + 42.3i)T^{2} \) |
| 67 | \( 1 + (-13.2 - 4.82i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (8.46 - 5.28i)T + (31.1 - 63.8i)T^{2} \) |
| 73 | \( 1 + (-2.68 - 5.50i)T + (-44.9 + 57.5i)T^{2} \) |
| 79 | \( 1 + (-13.7 + 1.93i)T + (75.9 - 21.7i)T^{2} \) |
| 83 | \( 1 + (3.68 - 0.783i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (0.234 + 1.32i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-8.28 - 10.5i)T + (-23.4 + 94.1i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.08726885994212120024353105906, −9.877866620396946745497002377969, −9.144616734066293055737685960317, −8.351233984318477182499471368755, −7.58319879243026712022413817114, −6.51876376093209280464638612058, −5.13477684823876635795489696446, −3.68588677072575985406987220120, −2.30300805066124144171493840245, −1.43331425836737268826877513085,
1.80679035074770291942890950123, 3.54396405259206119905005819822, 4.37287468289665637724338416713, 6.07722171412294207684785548566, 6.47954588530668769110749969556, 8.237197034953332788977037223786, 8.526008213212627254197366700953, 9.479679503750894197389150384722, 10.33981169026074064813275373784, 10.95403421973797537025430307406